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I was in dormitory when I started out as a vegetarian. It was a time when the food court didn't offered much vegetarian options: either the greasy egg and cheese on bagel, salad bar, a cold sub, smoothie, or patisserie (muffins, croissants, etc.) I didn't really mind that my food was cooked beside meats and stuff. I was just grateful that I get any type of vegetarian food offered.
The longer I stray away from the meat, I realise that my body became more sensitive to meat and reacted poorly if I had shared cookware with meat dishes. I can understand that people will want the veggie dishes be prepared with the separate cooking utensils, but it may or may not be feasible in every restaurants. I usual get to know the manager or the chef and pick the pure veggie sides or fries that they just use for cooking the specific dish to minimise the chances.
I do, however, tolerate more on permitting items such as prepackaged vegan cookies, vegan chocolate, etc. to be processed in the same facility as ones containing milk and egg. There is a growing amount of vegan options in South Africa, but nowhere as abundant as the United States. For now I will use the vegan options that's available and when I can find ones that are produced in a dairy-and-egg-free environment then I'll purchase them provided they are consistently good quality for the money I'm paying.
Shared equipment are not a problem for me, I know I do not contribute in the meat industry.
I do not mind sharing equipment. To me, making a fuss over shared equipment will only annoy others who prepare the food and give them the misconception that vegetarians/vegans are hard to accommodate. Many people I know don't even know what a "vegetarian" eats, let alone vegan!
My body is sensitive to meat, too, after more than a year being vegetarian. Whenever I accidentally consume meat (due to stocks, some chunks of meat in my dish when eating out despite requesting for mo meat added), my body will protest. However, using separate equipment is not always feasible in all restaurants, especially mom-and-pop restaurants. I will say that the best bet is to patronise vegetarian/vegan restaurants, prepare your own food or just bear with facility-sharing.
As for prepackaged vegan items, I will tolerate with facility sharing too. Nevertheless, I stay away from processed food whenever possible because they are not good for my health. Here in Malaysia, getting vegetarian items are a breeze, but not vegan because most vegetarians here are lacto-vegetarians. Vegan items such as cookies and bread are not easily accessible and are quite costly as well.